Orbits of the dynamical system f(z) = z^2+c where c is taken from the Mandelbrot Set

Directions for Using the Applet Orbits
1. Choosing c = p+iq
The first step is to choose a value of c = p+iq. You may choose from 16 pre-selected values for p and q by clicking on one of the pairs p, q in the upper left corner of the applet window. It is recommended that you use this method until you learn how the applet works.
2. Seeing the orbit
For each p and q that you choose, an appropriate region in the z-plane is displayed on the applet window (to the right of the choice menu). Clicking on a point will display the orbit of that point under iteration of f(z) = z^2 + c, where c = p + iq. On the left side choice menu you will see that you have the option of displaying the first 1000, 2000, 3000, 4000 or 10,000 points in the orbit. The default is 1000 points. The points in the orbit are rendered as balls with decreasing diameters; that is, the ball with the largest diameter is the first ball in the orbit, etc. You may superimpose any number of orbits on top of each other by clicking on different points. To start a new picture, click on Clear Screen. If you want to use your own choice of p and q, type them in the appropriate box and make sure that the box labeled Your Parameters is checked. If you want to get a rough idea where in the z-plane your orbit lies, you may click on Draw axes and then Clear Screen. (For some values of p and q, the appropriate region does not intersect the x (real) and y (imaginary) axes and you will not see the axes.)
3. How to Select the Number of Colors
The 16 pre-selected values of p and q were chosen to illustrate a small sample of the amazing variety of behavior exhibited upon iteration of f(z) = z^2+c. Some of the values yield attracting fixed points, others attracting cycles of various periods, others repelling fixed points or cycles,
and one illustrates one type of behavior in a neighborhood of a neutral fixed point. For example, for the second choice, p = -0.67, q = 0.35, f has an attracting cycle of period 9. The orbit of a point that lies in the basin of attraction of this cycle will approach the cycle in such a way that every 9^th point in the orbit will approach the same point in the cycle. Choosing the number of colors to be 9 for this example will illustrate this phenomenon clearly.
If n colors are chosen, the initial point in the orbit will be colored with color 0, the next point with color 1, the next point with color 2, etc., until n points have been colored. The (n+1)st point in the orbit will be colored with color 0 again. If you suspect that, for your choice of p and q, there is an attracting cycle of period n, try using n colors to verify your
conjecture.
For some fixed points (attracting or repelling), the orbits will spiral into or away from the fixed point in such a way that the angle of rotation is a rational multiple of 2*pi and by using a particular number of colors you will see a pattern emerge. For example, in the first choice, p = -0.4125, q = -0.5317, try 5 colors. In some cases you can get an effect
resembling phyllotaxis by first plotting some orbits in one choice of n colors, and then superimposing orbits drawn with a different choice of n colors.
4. An Experiment
Notice that the 6^th, 7^th and 8^th choices for p and q are very close together. The 6^th choice yields a repelling fixed point, the 8^th choice an attracting fixed point, and the 7^th choice
(p = -0.3905, q = 0.5868) yields a neutral fixed point. For this neutral fixed point f is locally equivalent to a rotation and this fixed point is the center of a Siegel Disk [See 3]. Within this Siegel Disk f is equivalent to a rotation, and you can see the "circles" that are invariant under f.
Try entering your own values for p and q moving slightly away from the neutral fixed point. For example, try p = -0.395, q = 0.584 (first use 9 colors, then 13 colors, then 16 colors). Or try p = -0.395, q = 0.5875.
5. Using this Applet with the Mandelbrot Set and the Julia Sets Applets
Using the Mandelbrot Set Applet, you can enlarge a portion of the Mandelbrot Set and get the coordinates of points in the Set. You can choose a point c from one of the various bulbs, try to guess the period of the attracting cycle of f(z) for that value of c [See 1]. Then use that value of c = p+iq in the Orbits Applet to find out what the orbits of points near that cycle look like. You can also pick a point from the boundary of the Mandelbrot Set and then use that point in the Orbits Applet. In the Julia Set Applet you can look at the Julia Set for a particular c = p+iq, and then look at orbits of f(z)= z^2+c for that value of c.
6. If you find particularly interesting pictures
Send me the values of p and q, and the number of colors to use, and I will post them for other users.

Anne Burns, Department of Mathematics, Long Island University, C.W. Post Campus, Brookville, NY, 11548 aburns@liu.edu

References
1. Devaney, Robert L., A First Course in Chaotic Dynamical Systems, Addison-Wesley, 1998
2. Mandelbrot, Benoit B., The Fractal Geometry of Nature, W.H. Freeman and Co., 1983
3. Peitgen, H.-O. and P.H. Richter, The Beauty of Fractals, Springer-Verlag, 1986